GR:KvFs & Geodesics: Solving for L & Derivatives

[binbagsss]

Homework Statement

see below and question attached , parts c to e

Relevant Equations

see below

I am a bit confused with parts c,d,e . In general does ' write down ' the geodesics mean you can leave it in a form $d/ds( ..) =0$ for example here I get 3 equations given by :$\dot x^c =\frac{ k_c z^2}{R^2}$ (for$c =t,x,y$and k is a constant) , and let me use the lagrangian to replace the equation and then I get :$L=K -\frac{\dot{z}^2 R^2}{z^2}$ where L will be set$>0 <0$ or $=0$ depending on whether s-l , t- l etc and K is a sum of the constants $k_c ^2$

d) the above seems to leave me with a pretty trivial answer to this part which is partly why I'm wondering whether the above is correct or not , since $L <0$ and the second term is positive . ( I thought perhaps I may need to look at the $z$ geodesic equation more explicitly and maybe solve for $z(s)$ or replace it with$\dot{z}$ but my $z$ equation looks non linear and second order ... is the idea to solve for $z(s)$ here before making a deduction on the conditions for the conserved quantities ?e) again this question makes me question my part c ) since , this again seems pretty trivial :

I'm wondering what the difference between ' write down explicitly ' and ' write down ' . I would interpret explicit as perhaps solving the Euler Lagrange equations and taking the second derivative wrt $s$ etc . But , using this method for all four equations , the value of $L$ is not specified - I.e which type of geodesic . Whereas if I use three Euler Lagrange equations and replace one of them with the lagrangian ( plugging in the other variables that have been solved for etc - Kvfs and constants ) I can specify the nature of the geodesic via the value of $L$ . But then in this case c) and e) are pretty much the same - the only different is specifying the value of L and computing the derivative wrt $s$ for the constant equations via kvfs I would have wrote down for part c) .Many thanks , really appreciated.

 

[PAllen]

I would interpret "write down the geodesic equation for ..." as referring to a differential equation simplified maximally using symmetries and conserved quantities. For the null geodesics, I would think (without doing the exercise) that they expect an explicit formula in the stated coordinates, e.g. that for the exterior Schwarzschild metric in Schwarzschild coordinates, there is a family of spacelike geodesics defined by constant t and angular coordinates, with varying r. I would be expecting that null geodesics simplify to the extent of writing actual equation, rather than diff.eq.